On the dual graphs of Cohen–Macaulay algebras
Bruno Benedetti
∗Dept. Mathematik u. Informatik, FU Berlin [email protected]
Matteo Varbaro
∗∗Dip. Matematica, U. Genova [email protected]
October 17, 2014
Abstract
Given an equidimensional algebraic set X ⊂ Pn, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne’s connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne’s result:
(1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo–Mumford regularity of X.
(The bound is best possible. For coordinate arrangements, it yields an algebraic extension of Balinski’s theorem for simplicial polytopes.)
(2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in Pn, then the diameter of the graph G(X) is less than or equal to codimPnX.
(The bound is sharp; for coordinate arrangements, it yields an algebraic expan-sion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)
On the way to these results, we show that there exists a graph which is not the dual graph of any simplicial complex (no matter the dimension).
1
Introduction
Let I be an ideal in the polynomial ring S = K[x1, . . . , xn], where K is some field. For
simplicity, we assume throughout this paper that I is height-unmixed, that is, all minimal primes of I have the same height. The dual graph G(I) is then naturally defined as follows: First we draw vertices v1, . . . , vs, corresponding to the minimal prime ideals
{p1, . . . , ps} of I. Then we connect two vertices vi and vj with an edge if and only if
height I = height(pi+ pj) − 1.
∗Supported by the DFG Collaborative Research Center TRR109, “Discretization in Geometry and Dynamics”.
The dual graph need not be connected, as shown for example by the ideal I = (x, y) ∩ (z, w) inside C[x, y, z, w], whose dual graph consists of two disjoint vertices. The reader familiar with combinatorics should note that this ideal is monomial and squarefree, so via the Stanley–Reisner correspondence it can be viewed as a simplicial complex. There is already an established notion of “dual graph of a (pure) simplicial complex” and it is compatible with our definition, in the sense that if I∆ is the Stanley–Reisner ideal of a
pure complex ∆, the dual graphs of ∆ and of I∆are the same. This way it is usually easy
to produce examples of ideals with prescribed dual graphs. However, not all graphs are dual graphs of a simplicial complex, as we will see in Corollary 4.2.
Having connected dual graph is a property well studied in the literature under the name of “connectedness in codimension one”. Remarkably, it is shared by all Cohen– Macaulay algebras:
Theorem 1.1 (Hartshorne [Har62]). For any ideal I ⊂ S, if S/I is Cohen–Macaulay then G(I) is connected.
(It is well known that, if S/I is Cohen-Macaulay, then I is height-unmixed). But can we say more about how connected G(I) is, if we know more about I — for example, that I is generated in certain degrees, or that S/I is Gorenstein? This leads to the following question.
Problem 1.2. Give a quantitative version of Hartshorne’s connectedness theorem. There are at least two natural directions to explore: (a) lower bounds for the connec-tivity, and (b) upper bounds for the diameter.
Connectivity counts how many vertex-disjoint paths there are (at least) between two arbitrary points of the graph. Balinski’s theorem says that the graph of every d-polytope is d-connected. Since the dual graph of any d-polytope P is also the 1-skeleton of a d-polytope (namely, of the polar polytope P∗), an equivalent reformulation of Balinski’s theorem is “the dual graph of every d-polytope P is d-connected”. This was later extended by many authors, cf. e.g. [Bar82] [Ath09] [Wot09] [BV13]. Here is one extension due to Klee:
Theorem 1.3 (Klee [Kle75]). Let I be the Stanley–Reisner ideal of a d-dimensional tri-angulated homology manifold (or more generally, of any d-dimensional normal pseudo-manifold without boundary). The dual graph of I is (d + 1)-connected.
Stanley–Reisner rings of homology spheres are particular examples of Gorenstein rings, so one can ask whether S/I Gorenstein implies that G(I) is highly connected. The answer is negative: As we show in Example 3.4, there are complete intersection ideals I such that G(I) is not even 2-connected, because it has a leaf.
Nevertheless, it is indeed possible to “compromise” between Hartshorne’s theorem and Balinski and Klee’s results. Recall that a radical ideal is said to define a subspace arrangement if it is a finite intersection of (prime) ideals generated by linear forms. Main Theorem 1 (Theorem 3.8). Let I ⊂ S be an ideal defining a subspace arrangement. If S/I is Gorenstein and has Castelnuovo-Mumford regularity r, then G(I) is r-connected.
The Stanley-Reisner ring of a simplicial (homology) d-sphere has Castelnuovo–Mum-ford regularity d + 1. So Main Theorem 1 does imply that the dual graph of every (homology) d-sphere is (d + 1)-connected. However, Main Theorem 1 is much more general. In fact, the arrangements corresponding to squarefree monomial ideals are called coordinate. Let L be the class of all subspace arrangements obtainable from coordinate ones via linear changes of variables or via hyperplane sections; let P be the class of subspace arrangements whose defining ideal is generated by a product of variables. It is well known that
{coordinate subspace arrangements} ⊂ L ⊂ P ⊂ {all subspace arrangements}, and most subspace arrangements are not in P, as explained in [BPS05].
Our proof of Main Theorem 1 uses liaison theory, cf. [Mig98], and a homological result by Derksen–Sidman [DS02]. The bound is best possible, in the sense that:
(1) The conclusion “r-connected” cannot be replaced by “(r + 1)-connected” in general, cf. Example 3.13.
(2) The assumption “S/I Gorenstein” cannot be weakened, for example, to “S/I Cohen– Macaulay”: See Remark 3.9.
(3) Without assuming that I defines a subspace arrangement, the best one can prove is that G(I) is 2-connected, provided the quotient of S by any primary component of I is Cohen–Macaulay (Corollary 3.2). Without the latter assumption, one can infer nothing more than the connectedness of G(I), even if I is a complete intersection. Compare Example 3.4.
(4) Non-radical complete intersections whose radical defines a subspace arrangement, might have a path as dual graph, even if the regularity of S/I is very high: See Example 5.10.
The other direction in which Hartshorne’s theorem could be extended, is by estimating the diameter. Recall that the diameter of a graph is defined as the maximal distance of two of its vertices; so connectedness is the same as having finite diameter. But is there a sharp bound on diam G(I) depending only on the degree of the generators of I, say?
One result of this type has been recently found in the case of (squarefree) monomial ideals, using ideas from metric geometry.
Theorem 1.4 (Adiprasito–Benedetti [AB13], cf. Section 2.3). Let I ⊂ S be a monomial ideal generated in degree 2. If S/I is Cohen–Macaulay, then diam G(I) ≤ height I.
Beyond the world of monomial ideals, however, the situation is much less clear. From now on, we will call Hirsch the ideals I such that diam G(I) ≤ height I. The name is inspired by a long-standing combinatorial problem, posed in 1957 by Warren Hirsch and recently solved in the negative by Santos [San12], which can be stated as follows:
(Disproved) Conjecture 1.5 (Hirsch). If ∆ is the boundary of a convex polytope, then I∆ is Hirsch.
The work by Santos and coauthors [MSW13] implies that for any k one can construct squarefree monomial ideals I = I(k) with S/I even Gorenstein, such that diam G(I) = 21k and height I = 20k (Example 5.2). However, these non-Hirsch ideals are generated in high degree. This motivated us to make the following conjecture:
Conjecture 1.6. Let I ⊂ S be an arbitrary ideal generated in degree 2. If S/I is Cohen– Macaulay, then I is Hirsch.
In Section 5, we show some partial argument in favor of Conjecture 1.6, proving it for all ideals of small height or regularity. A positive solution of Conjecture 1.6 would instantly imply also a polynomial upper bound (in terms of the number of variables) for ideals generated in higher degree: See Proposition 2.11.
Using techniques that are essentially combinatorial, although some algebraic geometry is required for the setup, in Section 4 we are able to obtain the following result:
Main Theorem 2 (Theorem 4.7). Let C ⊂ PN be an arrangement of projective lines such
that no three lines meet in the same point. If C is canonically embedded, then its defining ideal I is Hirsch, that is, the diameter of the graph G(I) is not larger than codimPNC.
“Canonically embedded” refers here to the technical requirement that the canonical sheaf ωC is isomorphic to the pull-back of the twisted structural sheaf OPN(1). This
con-dition is natural in order to produce embeddings that are quadratic and Cohen–Macaulay. (As a scheme, C can be embedded in several ways; the canonical embedding tends to be quadratic, while other embeddings may result in ideals generated in very high degree.)
The paper is structured as follows: we start with a “background” section, consisting of essentially known results and useful reductions. The reader already familiar with combinatorics and commutative algebra may skip to Sections 3 and 4, which form the core of the paper. Section 5 is finally a repertoire of interesting examples.
2
Background
2.1
Combinatorics: Graph Connectivity and Diameter
All graphs we consider have neither loops nor parallel edges. A graph G is called k-vertex-connected (or simply k-connected) if it has at least k + 1 vertices, and any two vertices of G are joined by at least k vertex-disjoint paths. So 1-connected is the same as connected. Similarly, G is called k-edge-connected if it has at least k + 1 vertices, and any two vertices of G are joined by at least k edge-disjoint paths. 1-edge-connected is the same as connected. Obviously k-vertex-connected implies k-edge-connected for all k. The converse is true only for k = 1: for example, two squares glued together at a vertex yield a 2-edge-connected graph that is not 2-connected. In any k-edge-connected graph, every vertex has degree at least k. The converse is false.
There is a well known characterization of the two notions of connectivity:
Theorem 2.1 (Menger). Let G be a graph on n vertices. Let 0 < k < n be an integer. (i) G is k-connected ⇐⇒ G cannot be disconnected by removing less than k vertices,
however chosen.
(ii) G is k-edge-connected ⇐⇒ G cannot be disconnected by removing less than k edges, however chosen.
For a direct proof of this, see [Diestel]; both (i) and (ii) are easy instances of Ford– Fulkerson’s “max-flow-min-cut theorem”, cf. [Bol98].
The distance of two vertices in a graph is the number of edges of a shortest path joining them. The diameter of a graph is the maximum of the distances between its vertices. As the intuition suggests, the more connected a graph is, the shorter its diameter:
Lemma 2.2 (folklore). Let G be a graph on s vertices having t edges; (a) if G is k-connected, then diam G ≤ b(s − 2)/kc + 1;
(b) if G is k-edge-connected, then diam G ≤ bt/kc.
Proof. We show item (a); item (b) is analogous. Let d be the diameter of G. If d ≤ 1 the claim is obvious. If d ≥ 2, choose two vertices x, y at distance d. By the connectivity assumption, there are k vertex-disjoint paths joining x and y. Each of these paths contains at least d − 1 vertices in its relative interior. Together with x and y, this yields a set of at least k(d − 1) + 2 vertices inside G. So k(d − 1) + 2 ≤ s, whence the conclusion follows because d = diam G is an integer.
For any connected graph G with s vertices, one has diam G ≤ s − 1, with equality if and only if G is a path. Since we are interested in upper bounds for the diameter, in the next section we review the known upper bounds on the number of vertices of G = G(I).
2.2
Commutative Algebra: The number of minimal primes
Throughout this section, S will denote the polynomial ring K[x1, . . . , xn]; I will be a
height-unmixed graded ideal of S; Min(I) will denote the set of minimal primes of I. To provide an upper bound for the number of vertices of G(I), let us recall a simple definition. If d is the Krull dimension of S/I there is a polynomial h ∈ Z[t], called the h-polynomial, such thatP
i∈N dimK(S/I)i ti = h(t)
(1−t)d. The integer e(S/I) = h(1) obtained
by evaluating the h-polynomial at 1 is called multiplicity1 of S/I. The multiplicity satisfies the following additive formula:
e(S/I) = X
p∈Min(I)
dimK(S/I)p · e(S/p). (1)
From (1) we see that e(S/I) is a sum of | Min(I)| positive integers. This implies the following:
Lemma 2.3. For any height-unmixed graded ideal I, the number of vertices of G(I) is at most e(S/I).
In case I is a radical ideal, we have I =T
p∈Min(I)pand ISp= pSp for all p ∈ Min(I).
In particular, dimK(S/I)p = 1, which allows us to simplify Equation (1) as follows:
e(S/I) = X
p∈Min(I)
e(S/p). (2)
In fact, if I has no embedded primes, (2) holds if and only if I is radical.
1The multiplicity is sometimes called degree in the literature. We refrain from this notation to avoid confusions with the degree of the polynomials generating I.
Remark 2.4. It is well known that e(S/p) = 1 if and only if p is generated by linear forms. So if I ⊂ S is an ideal defining a subspace arrangement,
e(S/I) = | Min(I)| = number of vertices of G(I).
Remark 2.4 suggests that the case of subspace arrangements is one of the most promis-ing for findpromis-ing examples of ideals with large diameter. For subspace arrangements, in fact, the graph G(I) has the largest possible number s of vertices – so the upper bound diam G(I) ≤ s − 1 becomes less restrictive.
To prove further upper bounds for the number of vertices of G(I), we need to recall a classical definition. Let
· · · → Fj → · · · → F0 → S/I → 0
be a minimal graded free resolution for the quotient S/I. The Castelnuovo–Mumford regularity reg(S/I) of S/I is the smallest integer r such that for each j, all minimal generators of Fj have degree ≤ r + j. The regularity does not change if we quotient out
by a regular element. It can be characterized using Grothendieck duality as follows: reg(S/I) = max{i + j : Hmi(S/I)j 6= 0}, (3)
where Hi
mstands for local cohomology with support in the maximal ideal m = (x1, . . . , xn).
This implies the following, well-known lemma (cf. [BH93, Theorem 4.4.3]):
Lemma 2.5. Let I be a graded ideal. Let h(t) be the h-polynomial of S/I. If S/I is Cohen–Macaulay, then deg(h) = reg(S/I).
Lemma 2.6. Let I ⊂ S be a height-unmixed graded ideal of height c. Let s be the number of vertices of G(I).
(i) If all minimal generators of I have degree ≤ k, then s ≤ kc.
(ii) If S/I is Cohen–Macaulay and has Castelnuovo-Mumford regularity r, then
s ≤ r X i=0 c + i − 1 i .
Proof. (i) By Lemma 2.3, it suffices to prove that e(S/I) ≤ kc. Since the Hilbert
function is preserved under field extensions, without loss of generality we may as-sume that K is infinite. Let us choose an S-regular sequence f1, . . . , fc of degree-k
polynomials such that J = (f1, . . . , fc) ⊂ I. Then e(S/I) ≤ e(S/J ) = kc.
(ii) As before, we may assume that K is infinite. A linear Artinian reduction of S/I (that is S/I mod out a linear system of parameters) will look like a K-vector subspace of A = K[x1, . . . , xc]
(x1, . . . , xc)r
. But then e(S/I) ≤ e(A) = Pr i=0
c+i−1 i .
2.3
Combinatorial commutative algebra: reduction to radicals
In this section we provide a quick overview on the situation for monomial ideals, showing that for such ideals the connectivity and diameter problems can be reduced to the radical
case and ultimately to the world of simplicial complexes, where we can exploit the recent results of [AB13]. We sketch the basic definitions, referring to [MS05, Chapter 1] for details.
Let n be a positive integer. A simplicial complex on n vertices is a finite collection ∆ of subsets of {1, . . . , n} (called faces) that is closed under taking subsets. The dimension of a face is its cardinality minus one. A facet is an inclusion-maximal face; “d-face” is short for “d-dimensional face” and “vertex” is short for “0-face”. The dimension of a simplicial complex is the largest dimension of a face in it. A simplicial complex is pure if all its facets have the same dimension. The dual graph of a pure simplicial complex ∆ is defined as follows: The graph vertices correspond to the facets of ∆, and two vertices are connected by an edge if and only if the corresponding facets share a face of dimension one less.
The Stanley–Reisner ideal I∆ of a simplicial complex ∆ with n vertices is the ideal
of K[x1, . . . , xn] defined by I∆ := (xi1· · · xir : {i1, . . . , ir} /∈ ∆) . By construction, I∆ is
generated by squarefree monomials. Conversely, every radical monomial ideal J is gen-erated by squarefree monomials and can be written as J = I∆ for a suitable simplicial
complex ∆. So “simplicial complexes on n vertices” are in bijection with “radical mono-mial ideals of S = K[x1, . . . , xn]”. Moreover, the minimal primes of I∆ can be described
combinatorially via the formula I∆=
\
F facet of ∆
(xi : i /∈ F ).
The height of an ideal generated by c distinct variables is c. In particular, if ∆ has n vertices and all its facets are d-dimensional, the height of any minimal prime of I∆ is
n − d − 1.
Lemma 2.7. If I∆ is the Stanley–Reisner ideal of a pure simplicial complex ∆, the dual
graph of ∆ is G(I∆).
Proof. Let F, F0 be two facets of ∆. F and F0 are adjacent in ∆ if and only if PF
and PF0 have the same monomial generators, except one; if and only if height(I∆) =
height(PF + PF0) − 1; if and only if PF and PF0 are adjacent in G(I∆).
A simplicial complex is called flag if the Stanley-Reisner ideal of the complex is gen-erated in degree two. A simplicial complex ∆ is called Cohen–Macaulay (over K) if K[x1, . . . , xn]/I∆ is Cohen–Macaulay. A simplicial complex is called strongly connected
if its dual graph is connected. The star of a face F in a simplicial complex C is the smallest subcomplex containing all faces of C that contain F . A simplicial complex is called normal if it is strongly connected, and so are the stars of all its faces. It is well known that Cohen–Macaulay complexes are normal.
Let ∆ be a d-dimensional simplicial complex, and let F, G be two adjacent d-simplices. Let v be the only vertex that belongs to F but not to G. When we move from F to G, we abandon the star of v in ∆. A path in the dual graph of ∆ is called non-revisiting if it never reenters the star of a vertex previously abandoned. It is easy to see that in a d-dimensional simplicial complex with n vertices, any non-revisiting dual path can be at most n − d − 1 steps long. These notions are interesting for our diameter problem because of the following recent result:
Theorem 2.8 (Adiprasito–Benedetti [AB13]). Let ∆ be a flag normal simplicial complex of dimension d and with n vertices. Then any two facets of ∆ can be connected via a non-revisiting path. In particular, the diameter of the dual graph of ∆ is ≤ n − d − 1.
The proof uses ideas of metric geometry applied to simplicial complexes. Below we present an algebraic consequence.
Lemma 2.9 (cf. [HTT05]). Let I be an ideal of S = K[x1, . . . , xn]. Let
√
I be the radical of I. If S/I is Cohen-Macaulay, S/√I need not be Cohen–Macaulay. However, if I is monomial and S/I is Cohen-Macaulay, so is S/√I.
Corollary 2.10. Let I be a monomial ideal such that S/I is Cohen-Macaulay. If I is generated in degree 2 (or more generally, if each minimal generator has a support of ≤ 2 variables), then diam G(I) ≤ height I.
Proof. Clearly, also√I is generated in degree at most 2; moreover, height√I = height I and G(√I) = G(I). Furthermore, S/√I is Cohen-Macaulay by Lemma 2.9. Since √I is radical and monomial, it is the Stanley–Reisner ring of some simplicial complex ∆. By the assumptions, ∆ is flag and Cohen-Macaulay, so in particular normal. Moreover, if ∆ has dimension d and n vertices, by Theorem 2.8 the dual graph of ∆ has diameter ≤ n − d − 1. Since height√I = n − d − 1, via Lemma 2.7 we conclude
diam G(I) = diam G(√I) ≤ n − d − 1 = height√I = height I.
2.4
Reduction to quadrics
Here we show that ideals generated in degree 2 play a special role in understanding dual graphs of Cohen-Macaulay projective algebraic objects. In fact, there is a classical algebraic procedure, named after Giuseppe Veronese, that allows us to associate any Cohen–Macaulay algebra with a Cohen–Macaulay quadratic algebra with the same dual graph.
Let k, d, n be positive integers. Let I be an ideal of S = K[x1, . . . , xn], generated in
degree ≤ k. Set R = S/I. Let u1, . . . , uN be a list of all monomials in S of degree d, with
N = n+d−1d . Consider the d-th Veronese rings S(d) =M i≥0 Sdi ⊂ S and R(d) = M i≥0 Rdi⊂ R.
If T is the polynomial ring K[y1, . . . , yN], we have natural surjections
T φd// //S(d) ψd // //R(d).
(Here φdis the map induced by yi 7→ ui, and ψdis the restriction to S(d) of the projection
from S to S/I.) If we set πd = ψd◦ φd, we can define
Vd(I) = Ker πd= Ker φd+ φ−1d (I ∩ S (d)).
Since Ker φd is generated by quadrics, Vd(I) is generated in degree ≤ max{2, dk/de}.
Furthermore, we have that the graphs G(I) and G(Vd(I)) are the same, since
as projective schemes. Finally, since R(d) is a direct summand of R and R is integral over R(d), then R(d) is Cohen-Macaulay whenever R is, by a theorem of Eagon and Hochster
[BH93, Theorem 6.4.5].
This allows us to show how Conjecture 1.6 has implications for the diameter of the dual graphs of all ideals, not only of those generated in degree 2.
Proposition 2.11. Suppose Conjecture 1.6 is true. Let I ⊂ S be an ideal generated in degree ≤ k. If S/I is Cohen–Macaulay, then
diam G(I) ≤ (n + b(k − 1)/4c)
dk/2e
dk/2e! .
Proof. With the notation above, set d = dk/2e and e = b(k − 1)/4c. Then Vd(I) is
quadratic and G(I) = G(Vd(I)). Furthermore, T /Vd(I) is Cohen–Macaulay, because S/I
is. Assuming Conjecture 1.6, we get
diam G(I) = diam G(Vd(I)) ≤ N =
n + d − 1 d = (n + d − 1) · · · n d! ≤ (n + e)d d! .
2.5
Reduction to projective curves
Here we show that under some extra technical assumption (satisfied by subspace arrange-ments, for example) Conjecture 1.6 can be further reduced to the case where I defines a projective curve. The geometric intuition is to intersect our algebraic object in Pn with a
hyperplane in general position, so that the intersection, viewed as an algebraic object in Pn−1, has the same dual graph as the starting object.
Throughout this section, we require K to be an infinite field (not necessarily alge-braically closed).
Lemma 2.12. Let I ⊂ S be a radical homogeneous ideal such that S/I is a d-dimensional Cohen–Macaulay ring, with d ≥ 3. If S/p is Cohen–Macaulay for all p ∈ Min(I), then there exists a radical homogeneous ideal I0 ⊂ S0
= K[x1, . . . , xn−1] such that
(i) S0/I0 has dimension d − 1,
(ii) for each i and j the graded Betti number βi,j(S0/I0) equals βi,j(S/I), and
(iii) G(I0) = G(I).
Furthermore, there is a bijection φ : Min(I) → Min(I0) such that for each i and j, βi,j(S0/φ(p)) = βi,j(S/p) for all p ∈ Min(I).
Proof. Set Min(I) = {p1, . . . , ps}, R = S/I and Ri = S/pi. By making a change of
coordinates we can assume that xn ∈ S is general, so we have that A = R/(xn) and
Ai = Ri/(xn) are (d − 1)-dimensional Cohen–Macaulay rings. Since d − 1 ≥ 2, we have
H0
m(Ai) = 0, where Hm0 denotes the 0-th local cohomology with support in the irrelevant
ideal m ⊂ S. Therefore, Bertini’s theorem tells us that Ai is a domain. This means that
p0i = pi+(xn)
(xn) is a prime ideal contained in S
0 = S/(x
n). By setting I0 = I+(x(xnn)) ⊂ S0 we
obtain
We have that height(p0i) = n − d and
height(p0i + p0j) = (
height(pi+ pj) if height(pi+ pj) < n,
height(pi+ pj) − 1 otherwise
Since d ≥ 3, we conclude that G(I0) = G(I).
Proposition 2.13. Let I ⊂ S be a quadratic ideal defining a subspace arrangement. Assume that S/I is a d-dimensional Cohen–Macaulay ring, with d ≥ 3. Then there exists a quadratic ideal I0 ⊂ S0
= K[x1, . . . , xn−1], defining another subspace arrangement, such
that S0/I0 is a (d − 1)-dimensional Cohen–Macaulay ring and G(I0) = G(I).
Proof. A minimal prime p of I is generated by linear forms, so clearly S/p is Cohen– Macaulay. Lemma 2.12 guarantees the existence of the ideal I0. To see that I0 defines a subspace arrangement, it is enough to prove that I0 is radical. This follows immediately from Bertini’s theorem and the fact that S/I is Cohen–Macaulay of dimension > 1.
The results above allow us to reduce Conjecture 1.6 to the 2-dimensional case.
Corollary 2.14. If Conjecture 1.6 holds when dim(S/I) = 2 (that is, when the scheme Proj(S/I) is a curve), then it also holds for all quadratic ideals I such that, for all p ∈ Min(I), S/p is Cohen–Macaulay.
Corollary 2.15. If Conjecture 1.6 holds when the scheme Proj(S/I) is a union of lines, then it holds whenever I is quadratic and defines a subspace arrangement.
3
Gorenstein algebras and r-connectivity
To deal with Gorenstein algebras, we need a tool from liaison theory. Recall that inside the polynomial ring S, two ideals I and I0 without common primary components, are called geometrically G-linked if S/(I ∩ I0) is Gorenstein. (This is stronger than algebraically G-linked, a property of pairs of ideals widely studied in the literature; cf. e.g. [Mig98].) Liaison theory easily implies the following result:
Proposition 3.1. Let I ⊂ S be an ideal such that S/I is Gorenstein. Let q be a primary component of I. Let v be the vertex of G(I) corresponding to the minimal prime p =√q of I. If S/q is Cohen-Macaulay, then either
(1) I is primary and G(I) consists only of v, or
(2) the deletion of v from G(I) yields a graph G0 that is connected. Proof. Let us write I =Ts
i=1qiwhere for all i = 1, . . . , s, qiis pi-primary. Up to relabeling,
we may assume q1 = q. If s = 1 then I is primary and case (1) is settled, so assume
s ≥ 2. The graph G(I) is on vertices v = v1, v2, . . . , vs corresponding to the pi’s. Note
that G(I) − v = G(J ), where J = q2∩ q3∩ . . . ∩ qs. Now, since J is geometrically linked
to q by a Gorenstein ideal and S/q is Cohen-Macaulay, it follows by the work of Schenzel [Sch82] that S/J is Cohen-Macaulay as well (see Migliore [Mig98, Theorem 5.3.1]). In particular G(J ) is connected.
Corollary 3.2. Let I ⊂ S be an ideal such that S/I is Gorenstein. If S/q is Cohen-Macaulay for any primary component q of I, then either G(I) is a point, or it is a segment, or it is a 2-connected graph. In any case,
diam G(I) ≤ e(S/I) 2 .
Proof. G(I) is connected, and by Proposition 3.1, the deletion of any vertex leaves G(I) connected. Let s be the number of vertices of G. By Lemma 2.2, diam G(I) ≤ s/2, whence we conclude via Lemma 2.3.
Corollary 3.3. Let I ⊂ S be an ideal defining a subspace arrangement. If S/I is Goren-stein, then either G(I) is a point, or it is a segment, or it is a 2-connected graph.
Our goal is now to strengthen the conclusion of Corollary 3.3. But first, the following examples show that one needs particular caution with the assumptions of Proposition 3.1 and its corollaries. First of all, the Cohen–Macaulayness assumption on S/q is necessary. Example 3.4. Let I = (x0x3 − x1x2, x21x3 − x0x22) ⊂ Q[x0, . . . , x3] = S. Since I is a
complete intersection, S/I is Gorenstein and Cohen–Macaulay. The prime decomposition of I can be computed with the software Macaulay2 [M2]:
√ I = I = (x0, x1) ∩ (x2, x3) ∩ (x1x2− x0x3, x32− x1x23, x0x22− x 2 1x3, x31− x 2 0x2).
The third ideal is the ideal of the projection of a rational normal curve of degree 4: C = {[t4, t3u, tu3, u4] : [t, u] ∈ P1} ⊂ P3.
The celebrity of such a quartic curve resides in the fact that it was studied in Hartshorne’s paper [Har79], where C was shown to be a set-theoretic complete intersection in positive characteristic. It is unknown whether the same holds in characteristic 0. However, the coordinate ring of C is not Cohen–Macaulay. It is easy to see that G(I) is simply a path of two edges, since the primes (x0, x1) and (x2, x3) are not connected by an edge. Hence
G(I) is 1-connected, but not 2-connected. In fact, removing the vertex corresponding to C disconnects the graph.
The ideal of Example 3.4 is radical. We stress that for non-radical ideals, Proposition 3.1 requires the Cohen-Macaulayness of S/q (where q is the p-primary ideal), and not of S/p. The next examples highlight why this distinction is important.
Example 3.5. Let I = (x2
4−x3x5, x3x4−x2x5, x2x3−x1x5, x1x2−x0x3) ⊂ C[x0, . . . , x5].
The ideal I is a complete intersection. Its minimal primes are
p1 = the prime defining the projective closure of the affine curve (t, t3, t4, t5, t6)
p2 = (x5, x4, x2, x0),
p3 = (x4, x3, x2, x1),
p4 = (x5, x4, x3, x2),
p5 = (x5, x4, x3, x1).
If S = C[x0, . . . , x5], clearly S/p4 is Cohen–Macaulay. Using Macaulay2 we computed
vertex adjacent to 2 is 4, so deleting 4 disconnects the graph. How do we reconcile this with Proposition 3.1? If we search for the p4-primary ideal in a primary decomposition
of I, this is not p4. It is instead
q4 = (x25, x4x5, x24− x3x5, x3x4− x2x5, x2x3− x1x5, x22− x0x5, x1x2− x0x3, x43)
and one can check that S/q4 is not Cohen–Macaulay.
Example 3.6. Let p be the prime homogeneous ideal in S = Z2[x1, . . . , x6] defining the
projective curve
(t5u + t4u2 + u6, t5u + t4u2+ t2u4+ u6, t5u + t4u2+ tu5, t6+ t3u3 + tu5,
t6+ t5u + t4u2 + t3u3+ t2u4+ u6, t6+ t5u + t3u3+ tu5+ u6) ⊂ P
5.
One can see with Macaulay2 that S/p is not Cohen-Macaulay and p is generated by the 8 quadratic polynomials a = x2 4+ x1x5+ x4x5 + x4x6+ x5x6, b = x2x3+ x3x4+ x1x5+ x3x6+ x4x6+ x5x6+ x26, c = x2x4+ x3x4+ x1x5+ x3x5+ x25+ x4x6+ x5x6, d = x2 2+ x1x4+ x3x4+ x2x5+ x3x5+ x4x5+ x25+ x1x6+ x2x6+ x3x6+ x4x6 + x5x6+ x26, e = x2 3+ x3x5+ x25+ x1x6+ x4x6, f = x1x3+ x1x4+ x1x5+ x2x5+ x25+ x1x6+ x2x6+ x3x6+ x4x6 + x5x6, g = x1x2+ x3x4+ x2x5+ x3x5+ x4x5+ x1x6+ x3x6 + x5x6, h = x2 1+ x1x5+ x4x5+ x25+ x2x6+ x4x6+ x26.
The ideal I1 = (a, c, f, g) is a complete intersection and has radical equal to p, so p is a
set-theoretic complete intersection. G(I1) consists of a single point.
The ideal I2 = (b, f, g, h) is a complete intersection whose radical is strictly contained
in p. The minimal primes of I2 are
p1 = p
p2 = (x6, x4+ x5, x2+ x5, x1),
p3 = (x6, x5, x3, x1),
p4 = (x5 + x6, x3+ x6, x2, x1+ x6),
p5 = (x4 + x5+ x6, x3+ x5, x2 + x5+ x6, x1+ x6)
Hence the graph G(I2) consists of the edges 12, 14, 15, 25, 34, 45. In particular, G(I2) has
diameter 3. Since 3 is a leaf (only 4 is adjacent to it), G(I2) is not 2-connected. As
in Example 3.5, S/p4 is Cohen–Macaulay, but S/q4 is not, where q4 is the p4-primary
component.
Finally, the ideal I3 = (c, f, g, h) is again a complete intersection with radical strictly
contained in p. The minimal primes of I3 are
p01 = p p02 = (x6, x4+ x5, x2+ x5, x1), p03 = (x6, x5, x4, x1), p04 = (x5+ x6, x3+ x6, x2, x1+ x6), p05 = (x3+ x4, x2+ x4+ x6, x1+ x4+ x5+ x6, x24 + x52+ x5x6+ x26), p06 = (x4+ x5+ x6, x3+ x5, x2+ x5+ x6, x1+ x6)
The graph G(I3) has edges 12, 14, 15, 16, 23, 25, 26, 45, 46, 56. Such a graph has
diameter 3 and is not 2-connected: The vertex 3 is adjacent only to 2. As above, S/p2 is
Cohen–Macaulay but S/q2 is not, where q2 is the p2-primary component.
Next we show that the conclusion “2-connected” of Proposition 3.1 is best possible. Example 3.7. Let J be the homogeneous ideal of S = Q[x0, ..., x4] given by
J = (−x1x2+ x0x3, −x22+ x1x3, −x1x3+ x0x4).
J is a complete intersection, hence in particular S/J is Gorenstein (of Castelnuovo– Mumford regularity 3). One of the minimal primes p1 of J is well known, as it defines
the rational normal curve
C = {[t4, t3u, t2u2, tu3, u4] : [t, u] ∈ P1} ⊂ P4.
The other primes are p2 = (x0, x1, x2), p3 = (x0, x2, x3), p4 = (x2, x3, x4). J is “almost”
radical: a primary decomposition of J is
J = p1∩ q2∩ p3∩ p4,
where q2 = (x0, x1, x22) is p2-primary. For each primary component q of J , S/q is a
Cohen-Macaulay (and even level) algebra. However, G(J ) is not the complete graph on 4 vertices, because the edge between p2 and p4 is missing. (All other edges are there, so
G(J ) is K4 minus an edge.) In particular, G(J ) is 2-connected, but not 3-connected: The
deletion of the vertices corresponding to p1 and p3 disconnects it.
With all these careful distinctions in mind, we are ready to announce our main result. Theorem 3.8. Let I ⊂ S be the defining ideal of a subspace arrangement. If S/I is Gorenstein of Castelnuovo–Mumford regularity r, then G(I) is r-connected.
Proof. Let K be the algebraic closure of K, S0 = S ⊗KK and I0 = IS0. Since S ,→ S0 is faithfully flat, we have that S0/I0 is Gorenstein and has regularity r. Furthermore, if I = p1 ∩ . . . ∩ ps, again, by the flatness we have I0 = p1S0 ∩ . . . ∩ psS0. Extensions of
prime ideals are not prime in general, but since our pi’s are generated by linear forms,
the piS0 are also prime ideals. So, I0 is the defining ideal of a subspace arrangement, and
G(I0) = G(I). This means there is no loss in assuming that K is algebraically closed. Let d = dim(S/I). By Lemma 2.12, we can assume that d = 2. This has the advantage that “connected in codimension one” is the same as “connected”. Let us write
I =
s
\
i=1
pi ⊂ S = K[x1, . . . , xn]
where the pi’s are ideals generated by linear forms and have height n − 2. For the rest of
the proof, for any subset A ⊂ {1, . . . , s} we set IA=
T
i∈Api.
To show that G(I) is r-connected, we must verify that G(IA) is connected for any
subset A ⊂ {1, . . . , s} such that |{1, . . . , s} \ A| < r. Notice that, because IA is radical
and K is algebraically closed, we have:
G(IA) is connected ⇐⇒ CA is connected ⇐⇒ H0(CA, OCA) ∼= K ⇐⇒ H
1
where CA is the curve Proj(S/IA) ⊂ Pn−1 and m is the irrelevant ideal of S.
Set B = {1, . . . , s} \ A, IB = ∩i∈Bpi and CB = Proj(S/IB). Then CA and CB are
geometrically linked by C = Proj(S/I), which is arithmetically Gorenstein. By Schenzel’s work [Sch82] (see also [Mig98, Theorem 5.3.1]) we have a graded isomorphism
Hm1(S/IA) ∼= Hm1(S/IB)∨(2 − r),
where −∨ means HomK(−, K). Therefore Hm1(S/IA)0 is nonzero if and only if there is
a nonzero map of K-vector spaces from Hm1(S/IB) to K of degree 2 − r, if and only if
H1
m(S/IB)r−2 6= 0. However, by the main result of Derksen and Sidman [DS02],
reg(S/IB) = reg(IB) − 1 ≤ |B| − 1 < r − 1,
so that H1
m(S/IB)j = 0 for all j ≥ r − 2 by Equation (3), and this concludes the proof.
Remark 3.9. It is natural to ask whether Theorem 3.8 can be extended from the general-ity of subspace arrangements, to arbitrary ideals. The answer is negative. In fact, Example 3.7 presents an ideal J such that S/J is Gorenstein and has Castelnuovo-Mumford regu-larity 3, yet G(J ) is not 3-connected. Another example would be given by the complete intersection I = (x24− x3x5, x1x4− x0x5, x2x3− x1x5, x1x2− x0x3): the graph G(I) is
2-but not 3-connected, while reg(S/I) = 4.
Similarly, one could ask whether Theorem 3.8 can be extended from Gorenstein to Cohen–Macaulay subspace arrangements. The answer is once again negative, already for coordinate subspace arrangements. For example, let ∆ be the graph 12, 13, 23, 14, 45. The Stanley–Reisner ring K[x1, . . . , x5]/I∆ is Cohen–Macaulay of regularity 2. However,
G(I∆) is connected, but not 2-connected.
Corollary 3.10 (Klee [Kle75]). Let I = I∆ be the Stanley-Reisner ideal of a homology
d-sphere ∆. Then G(I) is (d + 1)-connected.
Proof. By Hochster’s formula [MS05, Corollary 5.12], if ∆ is a homology d-sphere, then its Stanley–Reisner ring is Gorenstein of regularity d + 1.
Corollary 3.11 (Balinski). If P is any (simple) d-dimensional convex polytope, the 1-skeleton of P is d-connected.
Proof. The 1-skeleton of P is the dual graph of the simplicial d-sphere ∆ = ∂P∗, where P∗ is the polytope polar dual to P . By Corollary 3.10, we conclude.
Corollary 3.12. Let I be a complete intersection of height c defining a subspace arrange-ment, and let d be the minimal degree of a generator of I. Then G(I) is (d−1)c-connected. Proof. If I = (f1, ..., fc), then the Castelnuovo-Mumford regularity of S/I is deg(f1) +
... + deg(fc) − c ≥ (d − 1)c.
It is easy to see that the connectivity bounds given by Theorem 3.8 and Corollaries 3.10 and 3.12, cannot be improved in general:
Example 3.13. Let Ir = (x1x2, x3x4, . . . , x2r−1x2r) ⊂ K[x1, . . . , x2r] = S. This Ir
is the Stanley-Reisner ring of the boundary of the r-dimensional crosspolytope. Since height(Ir) = r, the ideal Ir is a complete intersection. Moreover, the regularity of S/Ir
is exactly r. By Lemma 2.7, G(Ir) is the dual graph of the r-crosspolytope, or in other
words, the 1-skeleton of the r-cube. So G(Ir) is r-connected. However, every vertex of
4
Arrangements of lines canonically embedded
Let C be an arrangement of projective lines. Consider the graph G(C) whose vertices correspond to the irreducible components of C, and such that two vertices are connected by an edge if and only if the intersection of the two corresponding irreducible components is nonempty. Once C is embedded in some PN, we have G(C) = G(I), where I is the ideal
defining C. Whether this defining ideal I is quadratic or not depends on the embedding; and the same is true for whether S/I is Cohen-Macaulay. In this section, we will prove bounds on diam G(I) for a certain, special embedding of C, called “canonical embedding”. Such an embedding does not always exist, but when it does, it tends to produce defining ideals that are both quadratic and Cohen–Macaulay.
Remark 4.1. There are graphs G which cannot be realized as dual graphs of arrange-ments of projective lines. For example, take the graph
G = {12, 13, 14, 15, 16, 23, 24, 26, 35, 36, 45, 46, 56},
(which is K6 minus two disjoint edges.) An arrangement C of projective lines such that
G(C) = G would consist of 6 projective lines r1, r2, r3, r4, r5 and r6. Let P be the point
r1∩ r2. The three lines r1, r2 and r3 are pairwise incident. So there are two cases: either
r3 passes through P , or not.
In the first case, r1, r2 and r3 are not coplanar, because r4 touches two of them but
not all of them. So r4 touches r1 and r2 in two points Q and R, respectively, which are
both different than P . Hence r1, r2 and r4 all belong to the same plane. The fifth line r5
cannot belong to such plane, because it does not intersect r2. But r5 meets both r1 and
r4. So r5 passes through the point Q = r1∩ r4 and intersects r3 in a further point S. The
five lines of the arrangement are then contained in the union of two planes, determined by the two triangles P QR and P QS; at the intersection of the two planes lies the line r1.
So we reached a contradiction, because there cannot be a sixth line r6, different than r1,
yet incident to all lines of the arrangement.
In the second case, i.e. if r3 does not pass through P , the three lines r1, r2 and r3
must belong to a common plane. The lines r4 ad r5 cannot belong to such plane, since r4
does not intersect r3, and r5 does not intersect r2. Hence r4 passes through P = r1∩ r2,
and r5 passes through Q0 := r1 ∩ r3. Set R0 := r2∩ r3 and S0 := r4∩ r5. As before, this
five-line arrangements determines two planes, intersecting at the line r1; so there cannot
be a sixth line incident to all five lines.
As a consequence, any graph containing the above G as an induced subgraph cannot be realized as the dual graph of arrangements of projective lines.
Analogously to the proof of Lemma 2.12, one can show that, for any pure simplicial complex ∆, an arrangement of projective lines C obtained by taking general hyperplane sections of the coordinate arrangement defined by I∆ satisfies G(C) = G(∆). Therefore
Remark 4.1 implies the following:
Corollary 4.2. Some graph is not the dual graph of any pure simplicial complex.
We now need some algebraic geometry notation; we refer the reader to the standard textbook by Hartshorne [Har77, Chapter II.7] for proofs and further details.
Given an invertible sheaf L on C, if C is a projective curve the K-vector space L(C) is finite. Let us consider a basis s0, . . . , sN of L(C). The elements of L(C) are called global
sections. By [Har77, Chapter II, Theorem 7.1], there is a unique morphism φ : C → PN
such that L is isomorphic to the pull-back φ∗(OPN(1)) and si = φ∗(xi), where the xi’s
are the coordinate functions on PN. In particular, L(C) is isomorphic as vector space to
S1, where S = K[x0, . . . , xN]. The sheaf L is called very ample if this morphism φ is an
immersion.
If P is an arbitrary point on the curve C, we denote by LP the stalk of L at P . By
mP we denote the maximal ideal of the local ring OC,P. For any global section s in L(C),
sP will denote the image of s in the stalk LP. The zero locus of s is
(s)0 = {P in C such that sP ∈ mPLP}.
With the notation above, one can prove the following well-known fact:
Lemma 4.3. If L is very ample, s is a global section of L and ` is the unique element of S1 such that φ∗(`) = s, then the points of (s)0 correspond to the points of intersection
between the curve C and the hyperplane defined by `.
A curve C is called locally Gorenstein if all the stalks OC,P, where P ranges over the
points of C, are Gorenstein rings.
Lemma 4.4. Any arrangement of projective lines is locally Gorenstein, provided no three lines of the arrangement meet in a common point.
Proof. If P belongs to one line only, then OC,P is even a regular ring. Otherwise OC,P
has Krull dimension 1 and embedding dimension 2. In particular, it is Gorenstein. On a locally Gorenstein curve C, one can define another invertible sheaf, called canon-ical sheaf and usually denoted by ωC. (It coincides with the dualizing sheaf defined in
[Har77, Chapter III, Section 7] for any projective scheme X. By definition of Gorenstein ring, the dualizing sheaf is invertible if and only if the scheme is locally Gorenstein.) The genus of the curve C is the dimension of the finite vector space ωC(C). The genus has a
particularly nice interpretation if C is an arrangement of projective lines.
Proposition 4.5 (Bayer–Eisenbud [BE91, Proposition 1.1]). Let C be an arrangement of projective lines. If no three lines of C meet at a common point, then the genus of C equals t − s + 1, where t (resp. s) is the number of edges (resp. vertices) of G = G(C).
When the canonical sheaf is very ample, it defines (as we saw for L) an immersion φ0 : C ,→ PN, which is usually called the canonical embedding. With slight abuse of notation, we use the expression “C canonically embedded” to denote the image φ0(C) ⊂ PN. It is well known that canonical embeddings play a central role in the theory of nonsingular curves: If the genus of the curve is at least 3, typically ωC is very ample and
the corresponding ideal is quadratic and Cohen-Macaulay (compare [Eis05, Chapter 9]). For the purposes of the present paper this is not interesting, since (connected) nonsingular curves are irreducible. However, a similar philosophy holds also for reducible curves (see [BE91]).
Lemma 4.6. Let C be an arrangement of projective lines, in which no three lines meet at a common point. If the canonical sheaf ωC is very ample, then G(C) is 3-edge-connected.
Proof. First of all, the existence of ωC is guaranteed by Lemma 4.4. By contradiction,
we can find two distinct edges in the graph G(C) whose removal disconnects it. Let P, Q be the two points on the curve C corresponding to these two edges. Let us consider the subspace of ωC(C)
W = {s ∈ ωC(C) such that (s)0 contains both P and Q}.
By [BE91, Proposition 2.3], W has codimension 1 in ωC(C). Now we use the assumption
that ωC is very ample, or in other words, that the morphism φ0 : C ,→ PN is an immersion.
Let V be the K-vector space formed by the linear forms of S = K[x0, . . . , xN] that vanish
on both P and Q. By Lemma 4.3, W is isomorphic as vector space to V . However, V has codimension 2 in S1. But S1 is isomorphic to ωC(C), in which W has codimension 1:
A contradiction.
Recall that a height-unmixed ideal I is Hirsch if the diameter of G(I) is ≤ height(I). Theorem 4.7. Let C ⊂ PN be an arrangement of lines no three of which meet at a
common point. If C is canonically embedded, then its defining ideal I is Hirsch.
Proof. First of all, notice that N = g − 1 where g is the genus of C. Let s (resp. t) be the number of vertices (resp. edges) of the graph G(C). The ideal I has height g − 2, where g is the genus of the curve. By Proposition 4.5, g = t − s + 1, and by Lemma 4.6 G is 3-edge-connected. In particular, every vertex of G lies in at least 3 edges and s ≥ 4, which implies 2t ≥ 3s. If s < 2t/3, then
height I = g − 2 = t − s − 1 > t/3 − 1,
which, since height I is an integer, implies height I ≥ bt/3c. Now Lemma 2.2 (b) implies diam G ≤ height I.
If 2t = 3s, then G is trivalent, that is: Each vertex of G lies in exactly 3 edges. A 3-edge connected trivalent graph is also 3-connected by [BE91, Lemma 2.6], so Lemma 2.2 (a) and the fact that s ≥ 4 let us conclude because:
height I = g − 2 = t − s − 1 = s/2 − 1 = (s − 2)/2 ≥ b(s − 2)/3c − 1.
5
Further examples of Hirsch and non-Hirsch ideals
In this section we prove the Hirsch property for a few cases, including all ideals of small height or regularity.
Proposition 5.1. The following homogeneous ideals of S = K[x1, . . . , xn] are Hirsch:
(i) prime ideals;
(ii) ideals corresponding to finite sets of points; (iii) ideals of height 1 (that is, hypersurfaces);
(v) height-unmixed ideals in a polynomial ring with n ≤ 3 variables. Proof. (i) G(I) is a single point.
(ii) In this case, G(I) is the complete graph on s vertices. So diam G(I) = 1 ≤ height I. (iii) For any two primes pi, pj of S, one has height(pi+ pj) ≤ height pi+ height pj. So if
height(I) = 1, for any two different minimal primes pi, pj of I we have height(pi) =
height(pj) = 1 and height(pi+ pj) = 2. So G(I) is the complete graph, as above.
(iv) Being G(I) connected, diam G(I) ≤ s − 1, where s is the number of vertices of G(I); but by Lemma 2.6, part (ii), we have s ≤ height(I) + 1.
(v) Let I ⊂ K[x1, x2, x3]. If the height of I is 1 resp. 2 resp. 3, we conclude by part (iii)
resp. (ii) resp. (i).
However, it is easy to find non-Hirsch ideals in a polynomial ring with four or more variables:
Example 5.2. The dual graph of the ideal
I = (x1, x2) ∩ (x2, x3) ∩ (x3, x4) ∩ (x4, x1+ x3)
is a path of three edges, hence has diameter 3. Since height(I) = 2, I is not Hirsch. Note that x1x3x4 is a minimal degree-3 generator for I, so I is not generated by
quadrics. In fact, height-2 (unmixed) ideals generated by quadrics are all Hirsch:
Proposition 5.3. Let I ⊂ S be a height-unmixed ideal of height c ≥ 2. If all the minimal generators of I have degree ≤ d and G(I) is connected, then diam G(I) ≤ dc− 2.
Proof. If d = 1, this is obvious, so we can assume d ≥ 2. Notice that, since G(I) is connected, I is height-unmixed. Therefore the number of vertices of G = G(I) is mostly dc by Lemma 2.6. So the only case in which the bound in the statement could fail is if G was a path on dc vertices. In such a case, however, I would be a complete intersection
of degree-k polynomials defining a subspace arrangement, so G would be c-connected by Corollary 3.12. We thus conclude by Lemma 2.2.
Corollary 5.4. Let I be a height-2 ideal, generated by quadrics. If S/I is Cohen– Macaulay, then I is Hirsch.
Proposition 5.5. If S/I is Gorenstein of regularity 2, then I is Hirsch.
Proof. If I contains linear forms, we can quotient them out without changing the regu-larity, so there is no loss in assuming I ⊂ m2.
Since Gorenstein implies Cohen–Macaulay, by Lemma 2.5 the h-polynomial of S/I has degree 2. Moreover, recall that if S/I is Gorenstein, then the h-polynomial is palyndromic. Set c = height(I); we have
e(S/I) = h(1) = h0+ h1+ h2 = 2 + h1 = 2 + c.
We distinguish two cases: either the number of vertices of G(I) is s ≤ e(S/I) − 1, or s = e(S/I). If s ≤ e(S/I) − 1, from the connectedness of G(I) we have
So, the only case left is when s = e(S/I), that is, when I defines a subspace arrangement. In this case, by Corollary 3.2 and Lemma 2.2 we obtain
diam G(I) ≤ s − 2 2 + 1 ≤ jc 2 k + 1 ≤ c.
In Proposition 5.5, note that I is quadratic unless it defines a hypersurface.
5.1
An ideal with many quadratic minimal primes
The intuition seems to suggests that, in dealing with Conjecture 1.6, the hardest case should be when I defines a subspace arrangement, because this is the case where G(I) has more vertices (cf. Remark 2.4). For this reason in the present paper we focused mostly on this case. However one can also find examples of quadratic complete intersections I such that Min(I) consists of many quadratic prime ideals. We study the graph G(I) in one such example, pointed out to us by Aldo Conca and Thomas Kahle, and prove it is anyway Hirsch.
Example 5.6. Let X = (xij) be an m×m- symmetric matrix (xij = xji) of indeterminates
over K, S = K[X] the corresponding polynomial ring in m+12 variables and I the ideal
generated by the principal 2-minors of X, namely
I = (xiixjj − x2ij : 1 ≤ i < j ≤ m).
The ideal I is a complete intersection of quadrics of height m2. Below, we are going to show that the graph G(I) has 2(m−12 ) vertices, and we will describe the corresponding
minimal prime ideals of I.
Notice that I is contained in the ideal I2(X) generated by all the 2-minors of X, which
is a prime ideal of the same height m2. Therefore I2(X) ∈ Min(I). We can find many
other minimal primes like this: If g is a change of variables of S, we denote gX = (g(xij))
Evidently the ideals I2(gX) ⊂ S have the same properties of I2(X): They are prime ideals
of height m2, S/I2(gX) is a Cohen–Macaulay ring of multiplicity 2m−1, and so on. Now,
let G be the set of changes of variables that fix the variables xii and change sign to some
xij’s with i < j. For any g ∈ G, we have I ⊂ I2(g(X)). Hence
{I2(gX) : g ∈ G} ⊂ Min(I).
We want to show that equality holds. Since the multiplicity of S/I is 2(m2), by the
additivity of the multiplicity it is enough to show that
|{I2(gX) : g ∈ G}| = 2( m−1 2 ). Certainly |{I2(gX) : g ∈ G}| ≤ 2( m−1 2 ), since |{I 2(gX) : g ∈ G}|e(S/I2(X)) ≤ e(S/I) by
different (notice that |G| = m2). To this end, for any subset A ⊂ {(i, j) : 1 ≤ i < j ≤ m} let us denote by gA the change of variables given by
gA(xij) =
(
xij if (i, j) /∈ A,
−xij if (i, j) ∈ A.
Now let us fix U = {(i, j) : 1 ≤ i < j − 1 ≤ m − 1}. The set U has cardinality m−12 and, if A and B are different subsets of U , one has I2(gAX) 6= I2(gBX). To see this, we can
assume that there is a j such that for some i, (i, j) ∈ A \ B. Pick the maximum index i doing the job, and notice that i ≤ m − 2 (since A is in U ). By denoting [a, b | c, d]gX the
2-minor of gX corresponding to the rows a, b and the columns c, d, we have: [i, i + 1 | i + 1, j]gAX = δxi,i+1xi+1,j + xi+1,i+1xi,j
[i, i + 1 | i + 1, j]gBX = δxi,i+1xi+1,j − xi+1,i+1xi,j,
where δ is −1 or +1 according to whether (i+1, j) does or does not belong to A. Therefore xi+1,i+1xi,j ∈ I2(gAX) + I2(gBX),
which means that I2(gAX) 6= I2(gBX). (Since it is a prime ideal, I2(gAX) does not
contain xi+1,i+1xi,j.)
Our next goal is to show that diam G(I) ≤ m−12 . To prove this, take two subsets A, B ⊂ {(i, j) : 1 ≤ i < j ≤ m} such that A ⊂ B and B \ A = {(i0, j0)}. We claim that
height(I2(gAX) + I2(gBX)) = height I + 1 =
m 2
+ 1.
In fact, it is easy to see that
I2(gAX) + I2(gBX) = I2(gAX) + (xi0,j0xij : both i 6= i0 and j 6= j0).
Consider the ideal I2(gAX) + I2(gBX) modulo I2(gAX), so that we get the ideal
J = (xi0,j0xij : both i 6= i0 and j 6= j0) ⊂ R = S/I2(gAX).
By Krull’s Hauptidealsatz, any minimal prime ideal p of (xi0,j0) has height at most 1, and
since p ⊇ J , it follows that height J ≤ 1. Because R is a domain and J is not the zero ideal, height J = 1. Thus the claim is proven.
Now, take two minimal prime ideals p and q of I. By what said before and the symmetry of the situation, we can assume that p = I2(X) and q = I2(gAX) for a subset
A of U = {(i, j) : 1 ≤ i < j − 1 ≤ m − 1}. Pick a saturated chain A1 ⊂ A2 ⊂ ... ⊂ Ak = A
such that |Ai| = i. Then, by what we proved above,
height(I2(X) + I2(gA1X)) = height(I2(gAi−1X) + I2(gAiX)) = 1 ∀ i = 2, . . . , k,
5.2
Cautionary examples and non-Hirsch ideals
Let us finish with some examples. The first one is a caveat concerning the “distance” between two minimal primes. In the monomial case, if three minimal primes p1, p2, p3 of a
monomial ideal I form a 2-edge path in G(I), then height(p1+ p3) is at most 2 + height p1.
Hence one is tempted to think that height(pi+ p1) should somehow measure the
graph-theoretical distance of pi from p1. This is false for non-monomial ideals, as the following
example (for n ≥ 4) outlines.
Example 5.7. Let S be the ring K[x1, . . . , xn, y1, . . . , yn]. Let px (resp. py) be the prime
ideal generated by x1, . . . , xn−1 (resp. by y1, . . . , yn−1). Clearly,
height px = n − 1 = height py.
Next, consider the 2 × n matrix with row vectors (x1, . . . , xn) and (y1, . . . , yn). Let p be
the prime ideal generated by the size-2 minors of such a matrix, and let I = p ∩ px∩ py.
It is well known that height p = n − 1. Moreover, p + px is contained in (x1, ..., xn), so
it has height n. It follows that in G(I) the primes p and px are connected by an edge.
Symmetrically, there is an edge between p and py. However,
height(px+ py) = height(x1, . . . , xn−1, y1, . . . , yn−1) = 2n − 2.
In conclusion, there is no upper bound for height(px+py), even if pxand py are two primes
at distance 2 in G(I).
Next, we highlight a construction (dual to taking products of polytopes) to obtain triangulated spheres whose Stanley–Reisner ring is “far from being Hirsch”. Recall that if P is any (convex) (d + 1)-dimensional simplicial polytope with n vertices, its polar dual Q is a (d + 1)-dimensional simple polytope with n facets: The graph of Q coincides with the dual graph of ∂P . Moreover, the k-fold product Qk = Q×. . .×Q is a k(d+1)-dimensional simple polytope with kn facets. If the graph of Q has diameter δ, it is not difficult to show that the graph of Qk has diameter kδ.
Example 5.8 (Matschke–Santos–Weibel). Matschke, Santos and Weibel [MSW13] re-cently constructed a simplicial polytope P with the following properties:
(i) The boundary ∆ = ∂P of P is a 19-dimensional sphere with 40 vertices; (ii) the dual graph of ∆ has diameter 21.
It follows that the ideal I∆ ⊂ K[x1, . . . , x40] has height 20 and diameter 21, so it is not
Hirsch. This is the smallest non-Hirsch sphere currently known. (The ideal I∆is monomial
and radical, but it is not generated in degree two. Moreover, S/I∆ is Gorenstein.)
Let us apply the dual product construction sketched before to the 20-dimensional polytope P above. If Q is the polar of P , let ∆k denote the boundary of the polar dual of
Qk. By construction, ∆
k is a simplicial sphere with 40k vertices and dimension 20k − 1.
Moreover, the dual graph of ∆k is just the graph of Qk, which has diameter 21k. If
Ik ⊂ K[x1, . . . , x40k] denotes the Stanley–Reisner ideal of ∆k, we have
Very recently, Santos produced d-dimensional simplicial complexes ∆ with diam G(I∆) ∈
nΘ(d) [San13, Corollary 2.12]. For Cohen–Macaulay d-complexes, however, the diameter
of the dual graph is bounded above by 2d−2n, which for fixed d is linear in n:
Theorem 5.9 (Larman [Lar70], see also [San13, Theorems 3.12 and 3.14]). Let I ⊂ S = K[x1, . . . , xn] be a (squarefree) monomial ideal of height c. If S/I is Cohen–Macaulay,
diam G(I) ≤ 2n−c−3n.
Our final example shows that even with the Cohen–Macaulay assumption, this type of upper bound (independent on the degree of generators) cannot exist outside the world of monomial ideals. In fact, even if we prescribe I to be a complete intersection, and even if we fix the parameters height(I) = 2 and n = 4, the diameter of G(I) can be arbitrarily high.
Example 5.10. If K is algebraically closed, for any N ∈ N, there are two polynomials f, g ∈ S = K[x1, . . . , x4] such that I = (f, g) is a complete intersection and diam G(I) =
N . To prove this, pick N + 2 linear forms `1, . . . , `N +2 ∈ S such that any 4 of them are
linearly independent, and set:
J = (`1, `2) ∩ (`2, `3) ∩ . . . ∩ (`N +1, `N +2).
By construction, J defines a connected union of lines in P3 and G(J ) is a path on N + 1
vertices. By a result of Mohan Kumar [Lyu89, Theorem 2.15], J is a set-theoretic complete intersection. In other words, there exist 2 polynomials f and g such that the ideals I = (f, g) and J have the same radical, namely, J . In particular, G(I) = G(J ) and diam G(I) = N . Note that
deg f + deg g = reg(S/I), and
deg f · deg g = e(S/I) ≥ e(S/J ) = N.
It follows that reg(S/I) ≥ 2√N . So if N is very large, the regularity of S/I is also large. In contrast, the graph G(I) = G(J ) is not even 2-connected. There is however no contradiction with Main Theorem 1. In fact, S/I is Gorenstein, but I does not define a subspace arrangement; whereas √I defines a subspace arrangement, but S/√I is not Gorenstein.
By Proposition 5.1, the phenomenon of Example 5.10 cannot appear in a polynomial ring S with less than 4 variables.
Acknowledgments. We thank David Eisenbud for suggesting us to look at his pa-per with Bayer [BE91], which inspired the results of Section 4. Thanks to Christos Athanasiadis, for pointing us the reference [Kle75]. Finally we would like to thank Aldo Conca for useful discussions and for proofreading part of the paper.
References
[AB13] K. Adiprasito, B. Benedetti, The Hirsch conjecture holds for normal flag complexes, to appear in Math. Oper. Research. Preprint at arxiv.org/abs/1303.3598, 2013. [Ath09] Ch. A. Athanasiadis, On the graph connectivity of skeleta of convex polytopes. Discr.
Comput. Geom. 42, pp. 155–165, 2009.
[Bar82] D. Barnette, Decomposition of homology manifolds and their graph, Isr. J. Math. 41, pp. 203–212, 1982.
[BE91] D. Bayer, D. Eisenbud, Graph curves. With an appendix by Sung Won Park, Adv. Math. 86, pp. 1-40, 1991.
[BPS05] A. Bj¨orner, I. Peeva, J. Sidman, Subspace arrangements defined by products of linear forms J. Lond. Math. Soc. 71, pp. 273–288, 2005.
[BV13] A. Bj¨orner, K. Vorwerk, On the connectivity of manifold graphs, to appear in Proc. Amer. Math. Soc. Preprint at arxiv.org/abs/1207.5381, 2013.
[Bol98] B. Bollob´as, Modern Graph Theory. Graduate Texts in Mathematics 184, Springer, 1998.
[BH93] W. Bruns, J. Herzog Cohen–Macaulay Rings. Cambridge University Press, 1993. [DS02] H. Derksen, J. Sidman, A sharp bound for the Castelnuovo-Mumford regularity of
sub-space arrangements, Adv. Math. 172, pp. 151–157, 2002. [Die05] H. Diestel, Graph Theory. Springer, 3rd ed., 2005.
[Eis05] D. Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics 229, Springer 2005.
[M2] D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at www.math.uiuc.edu/Macaulay2/.
[Har77] R. Hartshorne, Algebraic Geometry. Springer, Graduate Texts in Mathematics 52, 1977. [Har62] R. Hartshorne, Complete intersection and connectedness, Amer. J. Math. 84, pp.
497-508, 1962.
[Har79] R. Hartshorne, Complete Intersections in Characteristic p > 0, Amer. J. Math. 101, pp. 380-383, 1979.
[HTT05] J. Herzog, Y. Takayama, N. Terai, On the radical of a monomial ideal, Arch. der Math. 85, pp. 397-408, 2005.
[KK92] G. Kalai, D. J. Kleitman, A quasi-polynomial bound for the diameter of graphs of polyhedra, Bull. Amer. Math. Soc., pp. 315–316, 1992.
[Kal10] G. Kalai (coordinator), Polymath 3: Polynomial Hirsch Conjecture, online platform, September-October 2010. At gilkalai.wordpress.com/2010/09/29/ polymath-3-polynomial-hirsch-conjecture,
[Kle75] V. Klee, A d-pseudomanifold with f0 vertices has at least df0−(d−1)(d+2) d-simplices,
Houston J. Math. 1, 1975.
[Lar70] D. G. Larman, Paths on polytopes, Proc. London Math. Soc. 20:3, pp. 161–178, 1970. [Lyu89] G. Lyubeznik, A survey of problems and results on the number of defining equations,
MSRI Publications 15, pp. 375–390, 1989.
[MSW13] B. Matschke, F. Santos, C. Weibel, The width of 5-dimensional prismatoids. Preprint at arxiv.org/abs/1202.4701, 2013.
[MS05] E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Math-ematics 227, Springer, 2005.
[Mig98] J. C. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics 165, Birkh¨auser Boston, 1998.
[San12] F. Santos, A counterexample to the Hirsch conjecture, Ann. Math. (2) 176, pp. 383–412, 2012.
[San13] F. Santos, Recent progress on the combinatorial diameter of polytopes and simplicial complexes, TOP, Volume 21, Issue 3, pp. 426-460, 2013.
[Sch82] P. Schenzel, Notes on liaison and duality, J. Math. Kyoto Univ. 22, pp. 485-498, 1982. [Wot09] R. F. Wotzlaw, Incidence Graphs and Unneighborly Polytopes. PhD Thesis, TU Berlin, 2009. Available online at http://opus4.kobv.de/opus4-tuberlin/ frontdoor/index/index/docId/2116
